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A group L is a component of H if L is a quasisimple group which is a subnormal subgroup of H.

A prime factorization algorithm.

A group acts freely if there are no group fixed points. A point which is fixed by every group element would not be free to move.

A proper subgroup is a proper subset H of group elements of a group G that satisfies the four group requirements. "H is a proper subgroup of G" is written H subset G. The ...

The crystallographic point groups are the point groups in which translational periodicity is required (the so-called crystallography restriction). There are 32 such groups, ...

If a map f:G->G^' from a group G to a group G^' satisfies f(ab)=f(b)f(a) for all a,b in G, then f is said to be an antihomomorphism. Moreover, if G and G^' are isomorphic, ...

Let a group G have a group presentation G=<x_1,...,x_n|r_j(x_1,...,x_n),j in J> so that G=F/R, where F is the free group with basis {x_1,...,x_n} and R is the normal subgroup ...

A group G is said to be finitely generated if there exists a finite set of group generators for G.

A normalizer of a nontrivial Sylow p-subgroup of a group G.

The word "order" is used in a number of different ways in mathematics. Most commonly, it refers to the number of elements in (e.g., conjugacy class order, graph order, group ...